Two-dimensional backscattering Muellermatrix of sphere–cylinder birefringencemedia

E DuHonghui HeNan ZengYihong GuoRan LiaoYonghong HeHui Ma

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Two-dimensional backscattering Mueller matrixof sphere–cylinder birefringence media

E Du,a,b Honghui He,a Nan Zeng,a Yihong Guo,a,b Ran Liao,a Yonghong He,a and Hui Maa,baTsinghua University, Graduate School at Shenzhen, Shenzhen Key Laboratory for Minimally Invasive Medical Technologies,Shenzhen 518055, ChinabTsinghua University, Department of Physics, Beijing 100084, China

Abstract. We have developed a sphere–cylinder birefringence model (SCBM) for anisotropic media. The newmodel is based on a previously published sphere–cylinder scattering model (SCSM), but the spherical and cylind-rical scatterers are embedded in a linearly birefringent medium. A Monte Carlo simulation program for SCBM wasalso developed by adding a new module to the SCSM program to take into account the effects of birefringence.Simulations of the backscattering Mueller matrix demonstrate that SCBM results in better agreement with experi-mental results than SCSM and is more suitable to characterize fibrous tissues such as skeletal muscle. Using MonteCarlo simulations, we also examined the characteristics of two-dimensional backscattering Mueller matrix of SCBMand analyzed the influence of linear birefringence. © 2012 Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI: 10.1117/1

.JBO.17.12.126016]

Keywords: Mueller matrix; scattering; polarization; tissue; anisotropy.

Paper 12401 received Jun. 26, 2012; revised manuscript received Nov. 6, 2012; accepted for publication Nov. 19, 2012; publishedonline Dec. 12, 2012.

1 IntroductionIn the last few years, there has been increasing research interestin polarized light propagation in biological tissues. Most tissuesare fibrous. Characterization of such structurally anisotropic tur-bid media has been the subject of some recent works usingMonte Carlo simulation.1–4 In those studies, Mueller matrixserved as a powerful tool to obtain pathological information onbiological samples5–8 and evaluate the validity of tissue modelsand simulations. Antonelli et al.2 proposed a Monte Carlo modelincluding large and small spherical scatterers in a single layerabove the Lambertian substrate to explain the experimentalbackscattering Mueller matrix images of colon tissues. Woodet al.4 presented a Monte Carlo model for birefringent, opticallyactive, multiply scattering media to describe the propagation ofpolarized light in biological tissues. Wang and Wang9 proposeda sphere birefringence model (SBM) which contains sphericalparticles randomly suspended in linearly birefringent media.The scattering property of the medium is isotropic, and birefrin-gence is the only source of anisotropy. Recently, we proposed asphere–cylinder scattering model (SCSM) to characterize theanisotropic scattering property of fibrous tissues such as bulkskeletal muscles.10,11 In that model, the anisotropy of the fibroustissues is entirely attributed to the scattering of cylindrical scat-terers. However, for complicated tissues, both cylindrical scat-terers and the birefringence effect may coexist and contribute toanisotropic phenomena. In this paper, we present a new modelcontaining both contributions and confirm its validity by com-paring the simulations with experimental results.

We start from SCSM and construct the sphere–cylinder bire-fringence model (SCBM) by adding a birefringent mediumaround the spherical and cylindrical scatterers. The Monte Carlo

program for SCSM12 is also modified for SCBM. We use thenew program to simulate the backscattering Mueller matrix pat-terns of a sphere–cylinder birefringence medium and analyze theinfluence of birefringence on the patterns. Comparisons to theexperimental backscattering Mueller matrix patterns of skeletalmuscle show that SCBM results in better agreement than SCSMand SBM, which verifies the assumption on the two sources oftissue anisotropy and demonstrates that SCBM is a more generalmodel to characterize fibrous tissues such as skeletal muscle.

2 Method: Monte Carlo Simulation

2.1 SCBM for Anisotropic Media

As shown in Fig. 1, SCBM approximates the anisotropic turbidmedium to a mixture of solid spherical and infinitely longcylindrical scatterers embedded in a linearly birefringent med-ium. In this work, we assume that the surrounding medium is auniaxial material with its extraordinary axis being along thedirection of the cylinders.9 Parameters for the scatterers are thesame as those for the SCSM.11 They include number densitiesand sizes of both the spheres and cylinders, and the mean valueand standard deviation of the angular distribution for the cylin-ders. Parameters for the surrounding medium include the valueand axis direction of birefringence. Both the cylindrical orienta-tion and the axis direction of birefringence can be adjusted in thethree-dimensional (3-D) space. SCBM reduces to SCSM if thebirefringence is set to zero, or to SBM if the scattering coeffi-cient of the cylinders is set to zero.

2.2 Monte Carlo Algorithm

The Monte Carlo program for SCBM was developed based onthe polarization-sensitive SCSM program,12 which tracks thetrajectory and polarization state of photons. In the SCSMAddress all correspondence to: Hui Ma, Tsinghua University, Graduate School at

Shenzhen, Shenzhen Key Laboratory for Minimally Invasive Medical Technolo-gies, Shenzhen 518055, China. Tel: 86-755-26036238; Fax: 86-755-26036238;E-mail: mahui@tsinghua.edu.cn 0091-3286/2012/$25.00 © 2012 SPIE

Journal of Biomedical Optics 126016-1 December 2012 • Vol. 17(12)

Journal of Biomedical Optics 17(12), 126016 (December 2012)

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program, after each scattering event, the photon moves along anew direction determined by the Mie theory13,14 and may losepart of its energy because of the absorption. A statistical methodwas designed to determine whether the photon is scattered bythe cylinders or the spherical particles. The simulation processcontinues until the photon is completely absorbed or moves outof detection range. For SCBM, as the polarized photons transmitin the anisotropic medium, they alternately experience the trans-mission process in the birefringent medium and the scatteringprocess by the spherical or infinitely long cylindrical scatterers.

The Monte Carlo program for SCBM was also developedbased on the polarization-sensitive SCSM program12 and theSBM program,9 which describes the detailed algorithm of thebirefringence module. In SCBM, we assumed that the birefrin-gence effect is the property of the surrounding medium and doesnot affect the scattering phase function, but the birefringencedoes alter the polarization states of the photons as they propa-gate between two successive scattering events.9,15 We added anew module to the SCSM Monte Carlo program to calculatethe effects of birefringence.

In SCSM, the polarization states of the photons are altered byscattering only. If the medium around the scatterers is birefrin-gent, a scattered photon undergoes retardation between succes-sive scatterings. The new module calculates the changes ofStokes vector due to the birefringent medium:

S 0 ¼ Rð−βÞMðδÞRðβÞS; (1)

where MðδÞ is the Mueller matrix of the standard retarder andRðβÞ is the rotational matrix. MðδÞ can be expressed by

MðδÞ ¼

26641 0 0 0

0 1 0 0

0 0 cos δ sin δ0 0 − sin δ cos δ

3775; (2)

where δ is the retardation, which can be obtained by

δ ¼ 2πsn̄λ

Δn 0; (3)

where s is the transport path length between two successive scat-tering events, n̄ is the average refractive index, λ is the wave-length of light, and Δn 0 is the difference in refractive indicesexpressed by

Δn 0 ¼ n 0eðθÞ − no ¼

noneðn2o sin2 θ þ n2e cos2 θÞ1∕2

− no; (4)

where θ is the angle between the propagation direction of thephoton (u

⇀) and the extraordinary axis (→ e). The birefringence

value is defined by Δn ¼ ne − no, where ne and no are therefractive indices along the extraordinary and ordinary axes,respectively. RðβÞ can be expressed by

RðβÞ ¼

26641 0 0 0

0 cos 2β sin 2β 0

0 − sin 2β cos 2β 0

0 0 0 1

3775; (5)

where β is the rotational angle expressed as

β ¼ tan−1�−w · o 0

v · o 0

�; (6)

where o⇀ 0

is the projection of the ordinary axis onto the Stokesv⇀ − w

⇀plane.

2.3 Validity Test: Comparison with SBM

The SCBM becomes SBM if the scattering coefficient of cylin-ders is set to zero. To test the validity of the new module, we setthe scattering coefficient of the cylinders to zero in the SCBMprogram and compared the simulated Mueller matrix patternswith previously published results of SBM.16 Figure 2 showsthe simulation results of both models using the same set ofparameters for the spheres and birefringent medium.16 For thespheres, the diameter is 0.46 μm and the scattering coefficient is10 cm−1. The average refractive index of the birefringentsurrounding medium is 1.34, and its extraordinary axis isalong the horizontal x axis. The maximum retardation at thetransport length distance is calculated by ΔTR ¼ 2πΔnlTR∕λ

0 ¼2πΔn∕½λ0

μsð1 − gÞ�.16 In Fig. 2(a)–2(d), the birefringence valuetakes 0, 1.5 × 10−6, 7.5 × 10−6, and 4.5 × 10−5, corresponding

Fig. 1 Schematics of the sphere–cylinder birefringence model (SCBM).

Fig. 2 Monte Carlo–simulated backscattering Mueller matrices of theSBM: ΔTR ¼ 0 (a); ΔTR ¼ 0.1 (b); ΔTR ¼ 0.5 (c); and ΔTR ¼ 3 (d).Each image size is 1 × 1 cm. Note that the m11 images used thesame color maps but with the range from 0 to 1.

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to ΔTR ¼ 0, 0.1, 0.5, and 3, respectively. Figure 2 shows a goodagreement between the simulation results of SCBM and SBM.

2.4 Validity Test: Comparison with Mueller MatrixExperiments

2.4.1 Experiment setup

Figure 3 shows a typical experimental setup for backscatteringMueller matrix measurement.10 A polarized He-Ne laser (λ ¼633 nm) is used as the light source. A quarter-wave plate(QW1) and a linear polarizer P1 control the incident polarization.The incident light reaches the sample through a small hole at thecenter of the mirror (M), which is tilted at 45 deg to the incident

beam. The mirror directs the backscattered light from the sampletoward the camera through a quarter-wave plate (QW2) and a lin-ear polarizer (P2). The backscattered light is imaged by a 14-bitCCD camera (Canon APS-C). A laboratory reference frame isdefined such that both the muscle fiber orientation and the verticallinear polarization direction are along the y axis.

In the experiments, six different polarization states were usedfor both incidence and detection: horizontal linear (H), verticallinear (V), 45-deg linear (P), 135-deg linear (M), right circular(R), and left circular (L). A total of 36 reflectance images werecaptured with different combinations of the six incidence anddetection polarization states. The Mueller matrix was calculatedfrom these images:

M ¼

2664m11 m12 m13 m14

m21 m22 m23 m24

m31 m32 m33 m34

m41 m42 m43 m44

3775

¼ 1

2

2664HH þHV þ VH þ VV HH þHV − VH − VV PH þ PV −MP −MMHH −HV þ VH − VV HH −HV − VH þ VV PH − PV −MH þMVHP −HM þ VP − VM HP −HM − VPþ VM PP − PM −MPþMMHR − LLþ VR − RL HR − VRþ VL −HL PR −MRþML − PL

RH þ RV − LH − LVRH − RV − LH þ LVRP − RM − LPþ LMRR − RL − LRþ LL

3775:

(7)

For convenience, each reflectance image is indexed bytwo capital letters: the first letter represents the incidencepolarization state and the second letter represents thedetection polarization state. For example, HV representsa reflectance image of horizontally polarized (H) inci-dence and vertically polarized (V) detection.

2.4.2 Model sample for SCBM

In our previous work,10 we demonstrated that a sample withpolystyrene microspheres and well-aligned silk fibers immersedin water can be used as the model sample of SCSM. Meanwhile,other groups have reported of the model sample for SBM, whichincludes spherical scatterers embedded in a linearly birefrin-gence medium.4 The SBM sample consisted of polystyrenemicrospheres immersed in a polymer hydrogel called polyacry-lamide. During the experiments, the polyacrylamide sample wasstrained via extension to create birefringence along the directionof strain. In our sample developed for SCBM, scattering is pro-duced through the addition of polystyrene microspheres and silkfibers before the polymerization of the polyacrylamide, and

birefringence is produced through the straining of the polyacry-lamide. The fabrication process of the polyacrylamide has beendescribed in detail in the previous publication.4 In addition, thedirection of both the strain (the extraordinary axis of birefrin-gence) and direction of silk fibers, the scattering coefficient ofspherical scatterers, and the birefringence value can be adjustedin our sample.

In this paper, the sample is a three-layered medium consist-ing of a slab of well-aligned silk fibers along the y axis and poly-styrene microspheres submerged in the hydrogel. The first andthird layers are 5-mm-thick hydrogel containing 0.2-μm-diameter polystyrene microspheres (International Laboratory).The refractive indices of the microspheres and polyacrylamideare 1.59 and 1.39, respectively.4 The scattering coefficients are5 cm−1 determined by the concentration of the microspheres.The second layer contains only well-aligned silk fibers (pro-vided by Guangxi Institute of Supervision and Testing on Pro-duct Quality). The thickness of the silk fiber slab is 1 mm, andthe refractive index is 1.56. The diameter of the silk fiber, whichcontains a substructure, is taken as 1.5 μm, and the scatteringcoefficient of the silk layer is estimated as 65 cm−1 (Ref. 11).To discriminate the anisotropy due to the birefringence effect ofthe surrounding medium and the scattering by aligned cylinders,we aligned the silk fibers along the y axis and the direction ofstrain (the extraordinary axis of birefringence) along the 45-degangle axis on the x-y plane. The polyacrylamide sample wasstrained with the extension of 5 mm (the maximum extensionis 6 mm), and the difference in refractive indices was about1 × 10−5 (Ref. 4).

2.5 Results from the Experiment and SCBMSimulations

Using the new program based on SCBM, we simulated theexperimental results of the sample. All the parameters usedin the simulation matched the experiments. As shown in

Fig. 3 Scheme of experimental setup. Light source (LS): 12-mW He-Nelaser; P, polarizer; QW, quarter-wave plate; CCD, imaging camera.

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Fig. 4, the simulated result using SCBM agrees well with theexperimental result of the strained microsphere-silk polyacryla-mide. Although Fig. 4(a) and 4(b) are still different quantita-tively, the characteristic features in the two sets of Muellermatrix elements are very similar. The detailed analysis aboutthe effect of birefringence on the backscattering Mueller matrixis given in the following section.

3 Discussion

3.1 Comparison with the Simulations of SCSM

The characteristic features of the Mueller matrix of sphere scat-tering media have been reported previously by other groups.17,18

For the isotropic scattering media, the Mueller matrix has diag-onal symmetry properties. The m14 and m41 elements arealways blanks. The patterns of m11 and m44 are both circular.The m12 element is quatrefoil, and its intensity distributionsalong the x and y axes are nearly the same. The m22 elementhas a symmetric cross-like pattern with almost identical inten-sity distribution along the x and y axes. There are several pairs ofelements of similar intensity patterns but different rotations. Forexample, m13 can be obtained by rotating m12 by 45 deg, andthe same relation applies to the m31∕m21, m22∕m33, andm24∕m34 pairs.

In our previous paper,11 the Mueller matrix patterns of thesphere–cylinder scattering medium were described in detail.According to the Mie scattering theory for infinitely long cylin-ders,13,14 polarization-maintaining photons tend to be scatteredin the direction perpendicular to the cylindrical scatterers (i.e.,the x axis in Fig. 4). Therefore the intensity distributions of allthe Mueller matrix elements around the x axis are higher thanthose around the y axis. For example, m11 has the typical rhom-bus profile with an elongation along the x axis. m12 and m21have similar shapes of quatrefoils, but the intensity along the xaxis is higher than that along the y axis. The total intensity dis-tributions of them13 andm31 elements are weaker than those ofthe m12 and m21 elements. The m22 element still shows across-like pattern, but the intensity along the x axis is higherthan that along the y axis. The total intensity distributionand size of the m33 element are weaker than those of them22 element.

Compared to the sphere–cylinder scattering medium, thecharacteristic features of the Mueller matrix elements for thesphere–cylinder birefringence medium are very different. MonteCarlo–simulated backscattering Mueller matrix patterns of

SCSM and SCBM are shown in Fig. 5. The Mueller matrix ele-ments have been normalized to the m11 element to compensatefor the radial decay of intensity. The extraordinary axis of thebirefringent medium is along the y axis in the laboratory frame.For easier comparison, parameters of the scatterers are the sameas those in Ref. 11: The diameters of spheres and cylinders are0.2 and 1.5 μm, and their refractive indices are 1.59 and 1.56,respectively. The refractive index of the surrounding medium is1.33. The cylinders are aligned in the x-y plane, and their orien-tations fluctuate around the y axis following a Gaussian distri-bution with a standard deviation of 10 deg half width. Thescattering coefficients of spheres and cylinders are 10 and65 cm−1, respectively. In SCBM, since the birefringence valueof tissues ranges from 1 × 10−4 to 1 × 10−2 (Ref. 19), we setΔn to 0, 5 × 10−4, 1 × 10−3, and 1 × 10−2.

Figure 5 shows the influence of linear birefringence on thespatial intensity distributions of the backscattering Muellermatrix elements. The detailed features are summarized as

Fig. 4 The backscattering Mueller matrices of the strained microsphere-silk polyacrylamide with silk fibers aligned along the y axis and the direction ofstrain (the extraordinary axis of birefringence) along the 45-deg angle on the x-y plane (a) and the SCBM simulation (b). The size of each image is1 × 1 cm. Note that the m11 images used the same color maps but with the range from 0 to 1.

Fig. 5 Monte Carlo simulations of backscattering Mueller matrices.Sphere–cylinder scattering medium and a sphere–cylinder birefrin-gence medium (a) withΔn ¼ 5 × 10−4 (b); Δn ¼ 1 × 10−3 (c); and Δn ¼1 × 10−2 (d). The size of each image is 1 × 1 cm. Note that the m11images used the same color maps but with the range from 0 to 1.

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follows: (1) The m11 has the same typical rhombus profile asthat in SCSM. Linear birefringence does not affect the m11 ele-ment, which relates only to unpolarized light intensity. (2) Them12 and m21 elements have the same shapes of quatrefoils, andthe m22 also has the same cross-like pattern, as those in theSCSM simulations. For the present configuration, where boththe extraordinary axis and the cylindrical scatterers are alongthe vertical (y) direction,m12,m21, andm22 are hardly affectedby linear birefringence. This is because the Mueller matrix pat-terns represent mainly contributions from the polarization-main-taining photons. These elements relate only to the horizontal(0 deg) and vertical (90 deg) polarization components, whichare either parallel or perpendicular to the extraordinary axis andsubject to minimal effects due to the birefringent medium. (3)The intensity distributions of the m13, m23, m33, m31, m32,m33, and m44 elements are sensitive to birefringence Δn,since circular or 45-deg and 135-deg linear polarization statesare affected by retardation due to birefringence. As Δn in-creases, the m13, m31, m33, and m44 patterns stay in thesame shape but shrink in size. The shapes of the m23 and m32patterns change to quatrefoils stretched along the x axis. (4) Them14ðm41Þ), m24ðm42Þ, and m34ðm43Þ patterns have similarshapes to those ofm13,m23, andm33, respectively, but the con-trasts are different. The transformation between circular andlinear polarization states in the turbid medium is enhancedby linear birefringence.9

3.2 Effects of the Value and Orientation ofBirefringence on the Mueller Matrix Patterns

In Fig. 6, we compare the different responses ofm22 andm33 tochanges in the birefringence value Δn while keeping both theextraordinary axis and the cylindrical scatterers along the y

axis in the laboratory frame. As the value of birefringence Δnincreases, the total intensity of m22 remains almost the same,but the m33 intensity decreases rapidly and approaches zerowhen Δn exceeds 1 × 10−3. One may use m33 to detect the pre-sence of birefringence.

As Δn increases, the intensity of the m13, m23, m31, m32,and m44 Mueller matrix elements quickly decrease to zero, andthe terms related to 0-deg or 90-deg linear polarization (m12,m21) remain almost the same. These phenomena are consistentwith the simulation results of SBM (see Fig. 9 in Ref. 9), inwhich the degree of polarization (DOP) of diffusely reflectedlight does not change significantly with Δn when the incidentlight is in horizontal (0 deg) or vertical (90 deg) polarization, butdoes decrease fast and approaches zero with increasing Δn if theincident light is 45-deg linear polarization or circle polarization.The decrease of DOP means more recorded photons are depo-larized and the Mueller matrix loses their features. The intensityof the m14, m24, m34, m41, m42, and m43 elements will firstincrease then quickly decrease to zero as Δn increases fromzero. This is also consistent with the simulation results of SBM(see Fig. 4 in Ref. 16).

In the above simulations, the extraordinary axis of birefrin-gence has been set parallel to the orientation of the cylindricalscatterers, and to the y axis of the laboratory frame. We may alsoassume that the two directions are different and simulate theMueller matrix patterns for different orientation of the extraor-dinary axis. As shown in a previous work,11 the long axis of them11 pattern is always perpendicular to the direction of thefibers. For simplification, we set the direction of the cylindricalscatterers along the y axis in the laboratory frame and the extra-ordinary axis of birefringence in the x-y plane. All the patternsexcept for the m11 display asymmetric behavior, as shown inFig. 4(b). A further study indicates thatm22 is the most sensitiveto the angle mismatch. As shown in Fig. 7, we changed theextraordinary axis from the x axis to y axis at intervals of15 deg in the simulations while keeping the other parametersthe same as in Fig. 5(c). The m22 pattern has a cross-likeshape when the extraordinary axis is parallel or perpendicularto the cylinders, but is distorted when neither the fast axisnor the slow axis of birefringence matches the fiber orientation.Therefore, the m22 pattern may serve as an indicator for themismatch between the directions of the birefringence axis andthe fiber orientation. In real biological tissues, the birefringenceaxis may vary in the 3-D space. The corresponding features canalso be simulated using the current SCBM program.

3.3 Comparison with Experimental Results onSkeletal Muscles

In a previous paper,10 we demonstrated that SCSM can be usedto characterize the structural and optical properties of skeletalmuscle. The Mueller matrix elements of both skeletal musclesand a model sample containing polystyrene microspheres andsilk fibers show very similar features. However, more careful

Fig. 6 The total intensity of m22 and m33 with different birefringencevalueΔn. The extraordinary axis of the birefringence and the cylindricaldirection are both along the y axis in the laboratory coordinate. The dataare normalized by the maximum of m22.

Fig. 7 m22 patterns from sphere–cylinder birefringence media with the cylinders aligned along the y axis and the angle between the extraordinary axisand x axis of 0, 15, 30, 45, 60, 75, and 90 deg.

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examinations reveal inconsistency between the experimentallyobtained and the simulated results. For example, the third rowand the third column of the simulated Mueller matrix usingSCSM display clear patterns that are not apparent in the experi-mental results of skeletal muscle. Simulations using SCBMshow that this discrepancy is due to the birefringence effect.

In the present work, the skeletal muscle samples are freshbovine sternomandibularis muscle excised from the animals 2 hafter slaughtering. The Mueller matrix elements of the skeletalmuscle were compared with the simulations using SCBM,SCSM, and SBM as shown in Fig. 8.

The following parameters are used in the SCBM. The extra-ordinary axis of the birefringent surrounding medium and thedirection of the muscle fibers are both along the y axis. The bire-fringence value Δn is 2 × 10−3 (Ref. 20). The diameters of thespheres and infinitely long cylinders are 0.2 and 1.5 μm, respec-tively, and the refractive indices of scatterers are both 1.4. Theaverage refractive index of the surrounding medium is taken as1.33. The standard deviation of angular distribution of the cylin-ders is 10 deg. Using the previous published results10 as a refer-ence, the ratio of the scattering coefficients of spheres andcylinders is set at 1∕4 to give the best match between the shapesof the experimental and simulated images. The total scatteringcoefficients along the y axis is 50� 10 cm−1, which is close tothe scattering coefficient of the skeletal muscle.10 In the SCSM,the birefringence value is set to zero, and in the SBM, the scat-tering coefficient of cylinders is set to zero. The other para-meters are the same as those in the SCBM.

Compared with the experiments, simulations of the three ani-sotropic models (SCBM, SCSM, and SBM) had respective simi-larities and differences. The characteristic features in theMueller matrix patterns of anisotropic turbid media showed con-tributions by both optical anisotropy due to birefringence andscattering anisotropy due to cylindrical scatterers.

As shown in Fig. 8, for the Mueller matrix elements of a mus-cle sample, m11 has the typical rhombus shape, and m22 has across-like pattern with the dominant distribution along the xaxis. The simulations using SCBM and SCSM regenerated simi-lar features in these two elements. However, simulations usingSBM resulted in totally different shapes. Them11 pattern is cen-trosymmetric and m22 has a symmetric cross-like pattern withidentical intensity distributions along the x and y axes. Thus onehas to include cylindrical scatterers in the model to regeneratethese characteristic features in m11 and m22 of anisotropic tis-sues. These features of m11, m12, m21, and m22 are due to theanisotropy of the cylinders, since the polarization-maintainingphotons tend to be scattered to the direction perpendicular tothe cylindrical scatterers (the x axis).11

On the other hand, the third and fourth rows and columns ofthe experimentally obtained Mueller matrix for the muscle sam-ples show only weak patterns. Simulations using both SCBMand SBM regenerated such features, but simulations usingSCSM resulted in much clearer patterns, for example clear qua-trefoils for the m13 and m31 elements. More detailed simula-tions using SCBM showed that the birefringence effect isresponsible for smoothing out the distinctive features in thethird and fourth rows and columns of the Mueller matrixelements.

The above analysis proves that simulations using SCBMresult in better agreements with the experiments than SCSMand SBM. It should be pointed out that the Mueller matrixelements related to the third and fourth rows and columns donot show good agreements between the skeletal muscle experi-ments [Fig. 8(a) and also Ref. 7] and the SCBM simulations[Fig. 8(b)]. Simulation shows that these elements are all weakand noisy when birefringence exists, as shown in Fig. 8(b) and8(d). The corresponding experimental patterns also displaynoisy and irregular patterns,7 which may also contain measure-ment errors and inhomogeneity effects of the tissue sample.

One has to consider contributions from both the cylindricalscatterers and the birefringence effect to explain the character-istic features of polarized photons scattering in complicated ani-sotropic turbid media such as skeletal muscles. These two termson tissue anisotropy are perhaps corresponding to the anisotropyof form defined by the tissue microstructure and the anisotropyof material controlled by intrinsic anisotropic character of meta-bolic molecules.20

In addition, all the experimentally obtained m22 elementsshowed cross-like patterns without distortion. From the aboveanalysis in Sec. 3.2, we conclude that the extraordinary axisof the birefringent media is parallel to the direction of the fibrousstructures in all the samples we have measured, including theskeletal muscles. This may be a hint that the two come fromthe same origin.

4 ConclusionBased on the sphere–cylinder scattering model (SCSM), asphere–cylinder birefringence model (SCBM) was developed byadding birefringence to the medium surrounding the scatterers.The Monte Carlo program for SCSM was also revised forSCBM to take into account the birefringence effects. The char-acteristics due to birefringence were examined by comparisonsbetween the simulated backscattering Mueller matrix patternsusing both SCSM and SCBM. It was found that the thirdand fourth rows and columns of the Mueller matrix elements nolonger had distinctive patterns when the birefringence exceeded

Fig. 8 The backscattering Mueller matrices of fresh skeletal muscle (a),the SCBM simulation (b), the SCSM simulation (c), and the SBM simula-tion (d). The muscle fiber orientation, the extraordinary axis of the bire-fringent medium, and the cylinder orientation are all along the y axis.Note that the m11 images used the same color maps but with the rangefrom 0 to 1.

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Du et al.: Two-dimensional backscattering Mueller matrix of sphere–cylinder birefringence media

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a certain value. These matrix elements may serve as good indi-cators of strong birefringence. If the extraordinary axis is notparallel to the orientation of the cylinders and the y axis, allthe Mueller matrix elements except m11 will be distorted.These matrix elements can be used to detect the mismatchbetween the extraordinary axis and the orientation of the fibrousstructure. Finally we compared the experimental results of a ske-letal muscle sample with the simulations using the different tis-sue models, i.e., SCBM, SCSM, SBM, and the simulations fromthe SCBM can reveal the characteristic features of the Muellermatrix patterns of anisotropic tissues such as muscles better thanthose from SCSM and SBM. The good agreement between theexperiments and the simulations using SCBM confirms that forfibrous biological tissues, both optical anisotropy, such as bire-fringence, and scattering anisotropy contribute to the anisotropyof fibrous tissues. The experimental results also prove that theextraordinary axis of anisotropy and the orientation of fibrousstructure are always in parallel for the muscles, which mayhint that the two are interrelated.

AcknowledgmentsThis work has been supported by National Natural ScienceFoundation of China (NSFC) grants 10974114, 11174178,and 41106034 and Open Fund of Key Laboratory of Optoelec-tronic Information and Sensing Technologies of GuangdongHigher Education Institutes, Jinan University.

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Journal of Biomedical Optics 126016-7 December 2012 • Vol. 17(12)

Du et al.: Two-dimensional backscattering Mueller matrix of sphere–cylinder birefringence media

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